Representations of Solvable Lie Groups Basic Theory and Examples

The theory of unitary group representations began with finite groups, and blossomed in the twentieth century both as a natural abstraction of classical harmonic analysis, and as a tool for understanding various physical phenomena. Combining basic theory and new results, this monograph is a fresh and self-contained exposition of group representations and harmonic analysis on solvable Lie groups. Covering a range of topics from stratification methods for linear solvable actions in a finite-dimensional vector space, to complete proofs of essential elements of Mackey theory and a unified development of the main features of the orbit method for solvable Lie groups, the authors provide both well-known and new examples, with a focus on those relevant to contemporary applications. Clear explanations of the basic theory make this an invaluable reference guide for graduate students as well as researchers.

1. Basic theory of solvable Lie algebras and Lie groups; 2. Stratification of an orbit space; 3. Unitary representations; 4. Coadjoint orbits and polarizations; 5. Irreducible unitary representations; 6. Plancherel formula and related topics; List of notations; Bibliography; Index.

Didier Arnal is Emeritus Professor at the University of Burgundy and previously was Director of the Burgundy Mathematics Institute. He instituted and has worked over the past fifteen years on a cooperation project between France and Tunisia. He has authored papers on a diverse range of topics including deformation quantization, harmonic analysis, and algebraic structures.
Bradley Currey III is a professor at Saint Louis University (SLU), Missouri. Formerly the Director of Graduate Studies in Mathematics at SLU, he has also served as a co-organizer in the Mathematics Research Communities program of the American Mathematical Society. Much of his recent research has explored the interplay of the theory of solvable Lie groups and applied harmonic analysis.